Question

the instructions for the given integral have two parts, one for the trapezoidal rule and one for simpsons rule. complete the following parts.

\\( \int _ { 0 } ^ { 2 } \left( 3 t ^ { 3 } + 5 t \
ight) d t \\)

(simplify your answer.)

\\( \left| e _ { t } \
ight| = 0.75 \\)

(simplify your answer.)

c. use the formula \\( \left( \left| e _ { t } \
ight| / ( \text { true value } ) \
ight) \times 100 \\) to express \\( \left| e _ { t } \
ight| \\) as a percentage of the integrals true value.

3 %

(simplify your answer. round to the nearest integer as needed.)

ii. using simpsons rule complete the following.

a. estimate the integral with \\( n = 4 \\) steps and find an upper bound for \\( \left| e _ { s } \
ight| \\).

\\( s = \square \\)

(simplify your answer.)

Explanation:

Step1: Calculate the width of sub - intervals

The interval is \([a = 0,b = 2]\) and \(n = 4\). The width of each sub - interval \(\Delta x=\frac{b - a}{n}=\frac{2-0}{4}=0.5\).
The endpoints are \(x_0 = 0,x_1=0.5,x_2 = 1,x_3 = 1.5,x_4 = 2\).

Step2: Evaluate the function at endpoints

Let \(f(t)=3t^{3}+5t\).
\(f(x_0)=f(0)=3\times0^{3}+5\times0 = 0\)
\(f(x_1)=f(0.5)=3\times(0.5)^{3}+5\times(0.5)=3\times0.125 + 2.5=0.375+2.5 = 2.875\)
\(f(x_2)=f(1)=3\times1^{3}+5\times1=3 + 5=8\)
\(f(x_3)=f(1.5)=3\times(1.5)^{3}+5\times(1.5)=3\times3.375+7.5 = 10.125+7.5=17.625\)
\(f(x_4)=f(2)=3\times2^{3}+5\times2=3\times8 + 10=24 + 10=34\)

Step3: Apply Simpson's rule formula

Simpson's rule formula is \(S=\frac{\Delta x}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+f(x_4)]\)
Substitute \(\Delta x = 0.5\) and the function values:

$$
LATEXBLOCK0
$$

Step4: Find the fourth - derivative of the function

First, \(f(t)=3t^{3}+5t\)
\(f^{\prime}(t)=9t^{2}+5\)
\(f^{\prime\prime}(t)=18t\)
\(f^{\prime\prime\prime}(t)=18\)
\(f^{(4)}(t)=0\)

The error bound formula for Simpson's rule is \(|E_S|\leq\frac{(b - a)^5}{180n^{4}}\max_{t\in[a,b]}|f^{(4)}(t)|\)
Since \(|f^{(4)}(t)| = 0\) for all \(t\) (because \(f^{(4)}(t)=0\)), then \(|E_S| = 0\)

Answer:

\(S = 22\) and \(|E_S|=0\)